TUNING OF NONLINEAR MODEL PREDICTIVE CONTROL FOR QUADRUPLE TANK PROCESS

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1 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: TUNING OF NONLINEAR MODEL PREDICTIVE CONTROL FOR QUADRUPLE TANK PROCESS 1 P.SRINIVASARAO, DR.P.SUBBAIAH 1 Dr.M.G.R Educaional Research and Inuiion Universiy, Deparmen of ECE Sri Sunflower College of engineering, Deparmen of ECE 1 srinu.insru@gmail.com, subbaiah.nani45@gmail.com ABSTRACT Muli Inpu Muli Oupu (MIMO) ineracive process is more complex han Single Inpu Single Oupu (SISO) sysems. We consider wo inpus and wo oupus for four ank process, which is also called Quadruple Tank Process (QTP). I consiss of wo inpus as volages fed o moor pumps and waer levels in he lower anks are he wo oupus. The inpus are manipulaed by seing of valves. I has four ank ineracive loop operaion consising of wo lower anks and wo upper anks. The Model Predicive Conrol (MPC) echnique is more suiable and gives opimized operaional conrol in calculaing presen and fuure values of oupu. I unes oupu and inpus simulaneously o provide more sable (opimized) oupu wihin he permissible limis of olerance / error. MPC can sabilize all linear processes effecively and efficienly. I can also work wih nonlinear processes under exreme condiions. I offers an opimized resul under Predicive conrol P and Horizon conrol M, by uning P and M and is applicaion o QTP as a Nonlinear Model Predicive Conrol. Keywords: Nonlinear Model Predicive Conrol, Opimized, QTP, Predicive Conrol P, Horizon Conrol M 1. INTRODUCTION mainaining and operaing he sysem wihin he MPC echnique is more applicable for obaining admissible operaing region which is par of he opimized performance and is aracive as i offers boundary. feedback sraegy for linear processes. This same MPC echniques was inroduced and mehod can be applied o nonlinear sysems and implemened in indusries since 196 s. Frank obain equally good response or resul. This is Allgower e al [8] provided basic mahemaical referred o as moving horizon or receding horizon properies and formulaion of Nonlinear Model conrol. MPC mehods use linear or nonlinear Predicive Conrol. Jorge L. Garriage e al[5] models, o calculae presen and fuure values of provided opimized soluion of uning parameers dynamic sysems. Linear MPC heory is quie and applied various new uning mehods based on maure and has wide ranging applicaions from process response and analysis. Rahul Shridhar e al chemicals o aerospace indusries. Mos of he [6] delivered Sraegy of Unconsrain for muliple physical sysems are inherenly nonlinear in variable inpu and oupu for non linear model. exisence because of economical consrains and Leonidas G.Bleris e al [4] proposed ha qualiy of produc in process indusry. This requires 316

2 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: Opimizaion is par of MPC, where opimal conrols are inpus for a process for real ime and modeling sysem. Albero Beporad [7] discussed abou MPC conroller design for differen se poins for evaluaion of closed loop performance. one is for non linear model of QTP and he oher is a linear model of QTP [1], []. Linear MPC approaches he operaing poins by uning objecive funcion. Nonlinear MPC approaches he problem wihou consideraion o he operaing poin by uning of objecive funcion for nominal values. Pas Fuure Se poin (arge) This is applied for QTP wih opimized echnique for MPC of wo mehods. MPC mehod has wo ways of uning. The firs u y ŷ Pas oupu Prediced fuure oupu Conrol horizon, M u Predicion horizon, P mehod is based on simulaion of process model [11] or adjuss parameers based on process dynamics, which is approximaely adjusable. Similarly second one is by explicily deriving formulas considering various parameers of he process model wih respec o dynamics. The conroller design has o se predicion horizon P, conrol horizon M, weighs on he oupu Q, weighs on he rae of change of inpu λ, he reference rajecory parameer α and some consan parameers[5]-[7]. k - 1 k k + 1 k + k + M - 1 k + P Figure 1: Basic principle of Model Predicive Conrol K. H. Johansson [1] moivaed he implemenaion of four ank process wih an adjusable zero. Sivakumaran e al [] explained abou neural model predicive conrol for muli variable process. S.M. Mahadi Alvani e al [3] elaboraed feedback nework for QTP. This work focuses on he applicaion of MPC echniques o nonlinear model of QTP [1], []. The basic principle of MPC is as shown in fig 1. I depics he dynamic behavior of sysem over prediced horizon P and conrol horizon M and deermines he predefined open loop performance objecive funcion which has o be opimized. Performance measure is obained a ime, he conroller predics he fuure values by uning for P greaer han l. MPC has wo mehods of approach, These are uned by wo fundamenal conceps: Ac of uning is primary o develop appropriae parameers for process model. If he model exhibis poor performance, hen coninue uning ill he model saisfies he oupu requiremens. If model is accurae enough, hen res of uning is no essenial. Subsiuion beween robusness and performance is he second acion. Mos MPC users in indusry design eiher auomaic or deune conroller, so ha conrollers always say sable a various and differen operaing poins. This work is mainly focused upon modeling a four ank processor wih consan inpus from wo pumps however he response is sabilized hrough manipulaion of conrol valves. Here he inpus u1, u are se for consan flow rae bu he valve manipulaion is done irrespecive of he wo phases using MPC conroller, hrough differen uning mehods for obaining an opimized soluion. The 317

3 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: main emphasis is o une he parameers of nonlinear model Quadruple Tank Process hrough opimized NMPC. I provides for broad reamen of available mehod and also develops new mehods o une QTP. This paper is organized as follows: secion gives descripion of four ank process. The explained formulaion of MPC and sabiliy is explained in 3 & 4. Opimizaion and uning mehods are given in 5& 6.The analysis and simulaion resuls are given in secion 7. Finally he conclusion is given in 8.. DESCRIPTION OF FOUR TANK PRO- CESS Quadruple-ank process consiss of four inerconneced waer anks and wo pumps as shown in Figure. The arge is o conrol he level in he lower wo anks wih wo pumps. The process inpus are v1 and v (inpu volages o he pumps) y and he oupus are 1 = kch 1 y and = kch (volages from level measuremen devices). Mass balances and Bernoulli s law yield he following model [3], [4]: dh1 a1 a 3 γ 1k = gh gh3 + u d A 1 1 A1 A1 dh a a 4 γ k = gh + gh d A 4 + u A A ( γ ) dh3 a3 1 k = gh 3 + u d A 3 A3 ( γ ) dh4 a4 1 1 k1 = gh 4 + u d A 1 4 A4 (1) γ The parameers 1, γ (,1 ) are deermined from how he valves are se prior o an experimen. The acceleraion of graviy is denoed by g. The parameer values for he process are given in Table 1. TABLE 1: Parameer Process Values parameers Unis Values A1, A3 A, A 4 a1, a3 a, a4 cm cm cm cm k V c c m g c m s The QTP has minimum phase and non minimum phase condiion under mulivariable zero locaion a wo operaing poins based on wo phases. Normally i works eiher in minimum phase or non minimum phase a wo operaing poins based on he conrol valve seings while he flow rae disribuion is dependen on conrol valve seings alone. Where, γi is he flow disribuion o lower and diagonal upper ank, Ai is he cross-secion area, ai is he oule hole cross secion and h i is he waer level, in ank i respecively. The volage applied o pump i is u i and he corresponding flow is k i u i 318

4 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: Assumpion : The vecor field f (, ) = coninuous and saisfies condiion. (, ) f x u is a iniial Assumpion 3: Equaion has a unique coninuous soluion for any iniial condiion in he region of ineres and coninuous manipulaed inpu funcion u : [, M ] u and x prediced sae funcion : [, P] x coninuous Figure : Diagram for Quadruple Tank Process 3. FORMULATION OF NMPC Consider he non linear differenial equaion for sabilizing he problem [9]. = ( ) & x( ) = x () x f x, u,, y = g x u (3) u u,, x x, y y, (4) Where x R n u and m R deermine he vecor of saes and inpus. Denoes x and u are feasible se of inpus and saes and y is esimaed or measured oupu. We assume a se of feasible assumpions i.e., x and y as follows: Assume 1: In is simples form, u and x are given by consrains of he form u min u umax (5a) x min x xmax (5b) Real sysems and models are mainly used for predicing he fuure values wihin he limis seleced by he conroller. The finie horizonal open loop described above is mahemaically u formulaed as follows. inernal conroller (, ;, ) min J x u M P u ( ) is represened as + P J ( x, u ; P, M ) : = f ( x, u ) d + + M g ( u ) d Subjec o: = ( ) x x x f x, u τ, x, [, P] x (6) = (7a) + (7b) u, [, M ] u + (7c) ( τ ) = ( τ + ), [ M, P] u u P 4. STABILITY + + (7d) Comparing he prediced resul of open and closed loop behavior is always differen. An NMPC sraegy ha achieves sabiliy independen of he choice of performance measure, cos funcion and 319

5 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: consrains of model is desirable. We assume ha he predicion horizon and conrol horizon if se such ha P M, and P < M will resul in insabiliy [8]. The one way o achieve sabiliy is he use of an infinie horizon cos funcion, i.e., P in Equaion 6 is se a. Pracically as well as heoreically his may no deermine he response. More appropriae and feasible condiion is P > M. Similarly examine he model under non feasible condiions, where P = M, P < M. Whereas P = M exhibis somewha admissible response, exhibis a more aggressive response where P < M. The inpu compued as he soluion of NMPC opimizaion problem is equal o he closed loop rajecory of non-linear sysem a any given insance of ime. Basic seps for infinie horizon proof are based on use of value funcions [7], [8]. Feasibiliy a one sampling insance does impel for nex sampling insance for he normal case. 5. OPTIMIZED THEORY FOR MPC The designer needs o opimize conrol algorihm o minimize cos and maximize performance measure. These depend on he sysem variables, which are saes x, oupu y, racking error e and conrol u. Describe he process sae equaion of nonlinear ime invarian [1], [8] and [9] = x& f ( x, u ) (8) y = g( x, u ) (9) Performance funcion: f J ( x, u, y ) = wxyu ( x, y, u) d (1) Is minimized o he dynamic sysem which is represened as maximized performance [1] measure for deermining conrol law wih penaly h( x ( f )) erm f J = h( x ) + f ( x, u ) d (11) f =final ime, = iniial ime; f Opimal soluion o opimized problem is denoed u*() and repeaedly solved a sampling insans =kδ K=, 1,... for open loop conrol problem. Admissible opimal conrol law is defined for closed loop conrol for Equaion a sampling insans ( τ ) u * = u *, x, P, M, τ [, δ ] (1) The opimal value of NMPC open loop opimal conrol as a funcion of he sae will be denoed in he following as value funcion * V x, P, M = J x, u, P, M (13) In a similar mehod, we obain performance measure form, from equaions (11) o (13) (,,, ) ( ) J x u P M = h x P + + P + M f ( x ( ), u ( )) d + g ( u ( )) d se (14) The admissible conrols are consrained o lie in a U; i.e. u U. We firs approximae he coninuous operaion of equaion (8) by a discree sysem. ( + ) x x (, ) f x u (15) (, ) x + = x + f x u (16) 3

6 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: Shorening he above noion ( 1 ) ˆ, x k + = x k + f x k u k (17) ( + 1 ) ˆ, x k f x k u k (18) In a similar manner, we ge performance measure form as N 1 J = h( x ( k) ) + ( f ( x ( k ), u ( k )) + g ( u ( k) ) ) k= (19) To minimize he deviaion of he final sae of sysem from is desired values, here are more analyical squared erms much more analyically solvable han oher ypes. Because posiive & negaive deviaions are equally undesirable, so absolue value could be used in quadraic form. Using marix noaion: T N 1 T T J = x k Hx k + ( x ( k) Qx ( k) + u ( k) Rx ( k)) k= () P ( ) T J x k, u k, T, min( ( P T C = x k Qx k ) M T T + ( u k Ru k ) + x ( k) Hx ( k) ) (1) Opimized soluion for equaion (1) wih number of inervals ( ( / 1 ), ( / 1 ),, ) J x k k u k k P M = P T m in ( ( x ( k / k 1 ) Q x ( k / k 1 )) + M T ( u ( k / k 1 ) R u ( k / k 1 )) + T x k / k 1 H x k / k 1 ) ( ) ( ) coninuous funcion Q, H; R is real symmeric posiive semi-definie n n marix. Q is oupu weighed marix and R inpu weighed marix. H is soluion of Ricai equaion from linear sandard sae space equaion (3) ( + 1) = + = + x k A x k B u k y k C x k D u k 1 (3) T T T H = A HA A HB B HB + R BHA + Q (4) 6. TUNING METHODOLOGY OF NMPC MPC opimizaion is a funcion of inpu u and sae variable x; hese wo parameers are uned exernally by P, M and inernally by Q, λ, R as per equaion (). All hese parameers are however described underneah in he following secions. 6.1 Predicion Horizon P Differen oupus will be obained because of he inpu values of P as seling ime, rise ime is quie differen. Increasing he value of P ends o minimize conroller aggressiveness [6]. This secion provides various echniques o une he predicion horizon by Reference [5], [6]. The final horizon is se o be finie or infinie o ensure sabiliy. In his case, he final horizon is described based on uning resul for closed loop sabiliy of conrol sysem or process. Lis of guide lines are colleced from reference [5] P = N + M 1 (5) P = T 1 s (6) 8 + P = 9 T s (7) P = 9 5 T s (8) () Here if, k is presen, k-1 is pas, if k-1 is presen hen k is fuure. Here k is described as discree or η < P r Ts (9) P > M + d Ts (3) 31

7 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: All he parameers of any SISO or MIMO process are based upon he poles and zeros of ransfer funcion and no of oupus as well as inpus. Sabiliy, of course is a funcion of he ransfer funcion of process. Tuning he value of P affecs he sabiliy value so as o make he sysem reach he seady sae of he reference value, based on values of seling ime and rise ime. Normally he number of oupus and order are consan while he response sands differen for differen locaions of poles and zeros. Proposed new uning mehods for NMPC of P are elaboraed below. P = kη + r ηt s (31) ( η ) P in M+ C+ k ± 1 (3) < P < Tp k or η r (33) By defaul P = 1 is probable value of objecive funcion, as per sabiliy crierion P is uned from he various parameer, like, seling ime s, rise ime r, no of oupus k, higher order of process η, no of conrollers C, process response ime p, sampling ime TS, delay ime d and response of rise ime 6,8,9,95 w.r. p. P value is calculaed as average of number of oupus. 6. Conrol Horizon M Evaluaing he value of M, if i increases in value, i ends o become more aggressive over he predicion horizon (M>P). This is o monior and conrol he response of daa from oupu by adjusing he manipulaed variable. This leads o a rade-off beween increasing performance and robusness of formulaion of conrol law, as a defaul conrol horizon is equal o 1. Formulae conrol horizon wihou more aggressiveness and exising robusness of permissible compuaion load. Collecing uning mehods are from reference5 and exhibi he resul of he model. Implemen hose colleced from reference [5] as lised below. M = 6 T s (34) ( P ) M = in (35) 4 M is differen from P and opposiely working. Response of process depends on rise ime and seling ime. The value of M effecively unes inpus or manipulaed variable of equaion () and is inversely proporional o he rise ime and seling ime and is dependen on P as well. A high value of P minimizes he effec of M on he response. Apply new uning mehods based on he above equaions, which are designed mainly based on parameer seling ime s, rise ime r, number of oupus k, higher order of process η, sampling ime Ts. ( ) M = min in s,in P 4 ± 1 (36) M = in k n s (37) M k r M η s (38) (39) 6.3 Oupu weighed marix is represened by Q The oupu variables are relaively weighed according o heir significance in he process model. I provides individual significance relaive o oupu variable, wih he mos imporan variable having a larger weigh compared o ohers. Increasing linearly he weigh on he upper limi of oupu o achieve a smooh response ill he desired oupu is obained. The elemens of Q ha correspond o correced error have nonzero weigh o help in relaive predicion. Derived expression for he oupu weigh for minimum phase also works for non minimum phase for he closed loop; he bandwidh is made small enough as explained in he reference [5] below 3

8 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: T Q = C C (4) λ < 1 η P (44) Based on above expression and oupu weighs also consider smoohness, expression for boh non minimum and minimum phase will be Q < 1 (41) 6.6 Reference Trajecory parameers In MPC applicaion, reference rajecory provides he necessary pah o reach final desired se poin [1]. I can be specified in several differen ways. I is designed beween iniial value and final value Q T de C C (4) beween β j < 1, j=1 P. Here C is oupu marix of linear sae space equaion. 6.4 Weighs on he magniude of he inpus R In similar fashion, R allows o be weighed for inpu variable according o heir relaive imporance. R is normally considered as diagonal marix wih diagonal elemens of rm rm marix. I is referred as inpu weighing marix or move suppression marix. I is more convenien for uning parameers based on parameer of [6] and [1]. rij as suppression facor [5], 6.5 Weighs on he rae of change of inpus λ This secion discusses exising and new approaches for uning he weighs on he rae of change of inpus. Penalizing he rae of change produces a more robus conroller bu a he cos of he conroller becoming more sluggish. Small value adjusmens yield a more aggressive conroller. We consider uning guideline from reference [5], [6] as follows. λ < 1 mp (43) Based on he above approach and analyzing various guidelines, even a small change exhibis overshoos, bu decrease in rise ime and seling ime. This is compensaed by oupu weighs β = closedloop openloop (45) j s s < β j < 1 (46) 7. SIMULATION ANALYSIS We discussed exensively he applicaion of non linear processes o QTP for he lower wo anks. By giving appropriae weighs o he uning sysem we generae condiions ha are applicable for represenaion as for non linear differenial equaions. Response is ploed for sep inpu for differen uning condiions. Responses are calculaed from uning equaions for opimized soluions of NMPC for lower wo anks. I was observed ha he responses of differen uning mehods are based on differen condiions and uning parameers. Same rise ime and seling ime w.r. P, M, Q and λ are obained The responses of non linear processes on comparison found o exhibi almos same responses for small deviaion, because if P increases, a same ime M also increases however he value of P minimizes he effec of aggressiveness of M. If he value of M is kep consan while he values of oher parameers are varied slighly- he responses, 3, 4, 5 obained remain almos same as illusraed in fig 3. 33

9 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: Response 1 is defined by P=16, M=4,C=.5, Q=.9,λ=.315. Response is defined by P=9, M=,C=.5, Q=.8,λ=.554.Response 3 is defined by P=11, M=,C=.5, Q=.85,λ=.45. Response 4 is defined by P=1, M=,C=.5, Q=.67,λ=.5.Response 5 is defined by P=11, M=,C=.5, Q=.7,λ=.73. Figure 3: Response Of Lower Tanks H1, H Response 1 is calculaed for P from equaion (7), for M equaion (35), for Q from equaion (41), and λ from equaion (43). Response is calculaed for P from equaion (6), for M equaion (35), for Q from equaion (41), and λ from equaion (43). Response 3 is calculaed for P from equaion (8), for M equaion (35), for Q from equaion (41), and λ from equaion (43). Response 4 is calculaed for P from equaion (31), for M equaion (37), for Q from equaion (41), and λ from equaion (44). Response 5 is calculaed for P from equaion (3), for M equaion (37), for Q from equaion (41), and λ from equaion (44). Values for responses as follows and plos response as shown in figure 3, uning of inpu as shown in figure 4. Figure4: Tuning Of Inpu Parameers As per sabiliy crieria from secion (4), he way of response for NMPC of QTP is illusraed in figure 5. Resuls of simulaion provided very aggressive response as shown in fig 5 for unsable condiion also. Because of opimaliy for MPC can provide somewha aggressive response when compared wih remaining condiions. Examined he QTP for sabiliy condiions, P > M, P = M, and P < M is ploed for sep response. Response 1, & 3 is calculaed for P from equaion (3), M from equaion (37), Q from equaion (41), and λ from equaion (44). 34

10 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: Figure 6: Tuning of inpu values numbers Figure 5: Response of lower anks H1, H. Values for responses as follows and ploed response as shown in fig 5, uning of inpu is shown in fig 6. Response 1 is defined by P=11, M=,C=.5, Q=.7,λ=.73. Response is defined by P=11, M=11, C=.5, Q=.7, λ=.73. Response 3 is defined by P=, M=11, C=.5, Q=.7, λ= CONCLUSION This work provides elaborae uning mehods for Quadruple Tank Process hrough Nonlinear Model Predicive Conrol wih several condiions and consrains. We generaed differen responses wih minue deviaions for obaining seady sae condiions. Verified all possible sabiliy condiions based on conrol and prediced horizon. When conrol horizon is more han predicion horizon, i exhibis aggressive response. Proposed new uning mehods for NMPC, provides sable response. The usage of MPC regularly exhibis smooh response under all uning parameers. Because opimaliy of MPC can be provide somewha aggressiveness response for unsable comparing of remaining condiions. REFERENCES: [1] K. H. Johansson, The Quadruple-Tank Process: A Mulivariable Laboraory Process wih an Adjusable Zero, IEEE Transacions on Conrol 35

11 h Sepember 14. Vol. 67 No JATIT & LLS. All righs reserved. ISSN: E-ISSN: Sysems Technology, Vol. 8, No. 3, 4, pp [] N.Sivakumaran, V.Kirubakran and T.K.Radhakrishnan, Neural Model Predicive Conroller for mulivariable Process, IEEE Conference, 6, pp [3] S.M. Mahadi Alvani and Marin J. Hayes, Quaniaive Feedback Design for a Benchmark Quadruple Tank Process, ISSC, 6, pp [4] Leonidas G.Bleris and Mayuresh V. Kohare, Real Time Implemenaion of Model Predicive Conrol, American conrol conference, June 8-15, pp [5] Jorge L. Garriage and Masoud Soroush, Predicive Conrol Tuning Mehods: A Review, Ind. Eng. Chem. Res., Vol. 49, No. 8, 1, pp [6] Rahul Shridhar and Dogles J.Cooper, A Tuning Sraegy for Unconsrained Mulivariable Model Predicive Conrol, Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997, pp [7] Albero Beporad, Model Predicive Conrol Design: New Trends and Tools Proceedings, IEEE Conference, Decision &Conrol (U.S.A) 6. [8] Frank Allgower, Rolf Findeisen and Zolan K.Nagy, Non Linear Model Predecive Conrol: From Theory o Applicaion, Ind. Eng. Chem. Res., Vol. 35, No 3, 4, pp [9] Sergey Edward Lyshevski, Conrol sysem heory wih Engineering Applicaions, Jaico Publishers, Mumbai., 3. [1] Dale E. Seborg,Thomas F. Edgar and Duncan A.Mellichamp, Process Dynamics and Conrol, John Willy & Sons, Ld, New Delhi, 6. [11] B.Wayane Bequee., Process Conrol Modeling, Design and Simulaion, Prenice Hall, 3. [1] Donald E.Kirk., Opimal Conrol Theory: An inroducion, Dover Publicaion, N.Y,